# Homework Help: Help with Subgroups

1. Sep 21, 2010

### annoymage

1. The problem statement, all variables and given/known data

if H and K are arbitrary subgroup of G, prove that HK=KH iff HK is a subgroup of G

2. Relevant equations

n/a

3. The attempt at a solution

no problem to prove => direction

for <= i can prove KH is a subset of HK

only i got troubled to show HK ia subset of KH

x in HK

x=hk for some h in H ,k in K

i manipulate it many ways and always got the form x=khk for some h in H ,k in K

HELP, and sorry no latex ,i'm very buzy now ;P

2. Sep 22, 2010

### snipez90

Re: subgroups

Take b in HK so that b-1 = hk is in HK. How do you get b back?

3. Sep 22, 2010

### lanedance

Re: subgroups

so $k^{-1}h^{-1} \in KH[/tex] and [itex] hk \in HK$, what's its inverse?

4. Sep 22, 2010

### annoymage

Re: subgroups

aahhhh i see,

b in HK

b=hk for some h in H k in K

b^{-1} also is in HK

imply
$b^{-1}=(hk)^{-1}=k^{-1}h^{-1} \in KH$

right ??

5. Sep 22, 2010

### annoymage

Re: subgroups

wait wrong,

for any x in HK, x^-1 in HK so x^-1=hk for some h and k

then $x=(x^{-1})^{-1}=(hk)^{-1}=k^{-1}h^{-1} \in KH$

now this is correct right?

6. Sep 22, 2010

### lanedance

Re: subgroups

yeah that looks good